Optimal. Leaf size=70 \[ \frac{16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0118909, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {614, 613} \[ \frac{16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 614
Rule 613
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{(8 c) \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0285332, size = 57, normalized size = 0.81 \[ \frac{2 (b+2 c x) \left (4 c \left (3 a+2 c x^2\right )-b^2+8 b c x\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 78, normalized size = 1.1 \begin{align*}{\frac{32\,{c}^{3}{x}^{3}+48\,b{c}^{2}{x}^{2}+48\,a{c}^{2}x+12\,{b}^{2}cx+24\,abc-2\,{b}^{3}}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.88551, size = 406, normalized size = 5.8 \begin{align*} \frac{2 \,{\left (16 \, c^{3} x^{3} + 24 \, b c^{2} x^{2} - b^{3} + 12 \, a b c + 6 \,{\left (b^{2} c + 4 \, a c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1344, size = 194, normalized size = 2.77 \begin{align*} \frac{2 \,{\left (2 \,{\left (4 \,{\left (\frac{2 \, c^{3} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac{3 \, b c^{2}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac{3 \,{\left (b^{2} c + 4 \, a c^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac{b^{3} - 12 \, a b c}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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